Zero-flux approximations for multivariate quadrature-based moment methods

The evolution of polydisperse systems is governed by population balance equations. A group of efficient solution approaches are the moment methods, which do not solve for the number density function(NDF) directly but rather for a set of its moments. While this is computationally efficient, a specific challenge arises when describing the fluxes across a boundary in phase space for the disappearance of elements, the so-called zero-flux. The main difficulty is the missing NDF-information at the boundary, which most moment methods cannot provide. Relevant physical examples are evaporating droplets, soot oxidation or particle dissolution. In general, this issue can be solved by reconstructing the NDF close to the boundary. However, this was previously only achieved with univariate approaches, i.e. considering only a single internal variable. Many physical problems are insufficiently described by univariate population balance equations, e.g. droplets in sprays often require the temperature or the velocity to be internal coordinates in addition to the size. In this paper, we propose an algorithm, which provides an efficient multivariate approach to calculate the zero-fluxes. The algorithm employs the Extended Quadrature Method of Moments(EQMOM) with Beta and Gamma kernel density functions for the marginal NDF reconstruction and a polynomial or spline for the other conditional dimensions. This combination allows to reconstruct the entire multivariate NDF and based on this, expressions for the disappearance flux are derived. An algorithm is proposed for the whole moment inversion and reconstruction process. It is validated against a suite of test cases with increasing complexity. The influence of the number of kernel density functions and the configuration of the polynomials and splines on the accuracy is discussed. Finally, the associated computational costs are evaluated. (C) 2019 Elsevier Inc. All rights reserved.